Arithmetic Hecke operators as completely positive maps
Florin R˘adulescu
Abstract. We prove that the arithmetic Hecke operators are completely positive maps with respect to the Berezin’s quantization deformation product of functions on H/Γ. We then show that the associated subfactor defined by the Connes’ correspon-dence associated to the completely positive map has integer index and graph A∞.
The same construction for P SL(3, Z) gives a finite index subfactor of L(P SL(3, Z)) of infinite depth.
Les op´erateurs de Hecke arithm´etiques et l’alg`ebre de d´eformation de Berezin R´esum´e. On montre que les op´erateurs de Hecke arithm´etiques sont des applica-tions compl`etement positives par rapport `a la structure de l’alg`ebre d´efinie par la d´eformation quantique de Berezin, des fonctions sur le quotient H/Γ du demi-plan sup´erieur par un sous-groupe discret Γ de P SL(2, R). On montre ensuite que si Γ est P SL(2, Z), alors les sous-facteurs associ´es par la m´ethode des correspondances de Connes ont comme indice un nombre entier et le type A∞. La mˆeme
construc-tion pour P SL(3, Z) nous permet de construire un sous-facteur de L(P SL(3, Z)) d’indice entier et de profondeur infinie.
Version fran¸caise abr´eg´ee. Dans cet article nous montrons que les op´erateurs de Hecke op´erant sur le sous-espace des fonctions sur le demi-plan sup´erieur et invariantes par rapport au groupe P SL(2, Z), sont des applications compl`etement positives pour la structure multiplicative donn´ee par la d´eformation quantique,
Γ-invariante, de Berezin du demi plan sup´erieur. La structure de l’alg`ebre de d´eformation est d´etermin´ee par les calculs, (dans [GHJ], [Ra]), de la dimension de von Neumann pour les L(P SL(2, Z))-modules de Hilbert `a gauche, obtenus par la restriction au groupe P SL(2, Z) des repr´esentations unitaires de P SL(2, R).
Chaque op´erateur de Hecke est une somme d’applications compl`etement positives par rapport `a la structure de C∗
-alg`ebre de l’alg`ebre de d´eformation.
Par la m´ethode des correspondances de Connes, chaque application compl`e-tement positive d´etermine un sous-facteur. Nous calculons les invariants provenant de ces sous-facteurs par la construction de base de Jones pour les sous-facteurs. Nous montrerons que chaque sous-facteur a comme indice le carr´e d’un nombre entier.
Nous montrerons ansi que l’alg`ebre de fusion du sous-facteur est un quotient d’une sous-alg`ebre de l’alg`ebre de Hecke d’un ensemble de classes doubles. Le bimodule de Hilbert est simplement, dans ce cas, l’espace l2 associ´e `a une double
classe de Γ dans GL(2, Q).
Ce genre de correspondances d’indice fini peut ˆetre construit chaques fois qu’on s’est donn´e un sous-groupe presque normal d’un groupe discret. Par cons´equente, nous pensons qu’un autre exemple int´eressant pourrait ˆetre obtenu `a partir des sous-groupes presque normaux consid´er´es dans [BC]. On a ci-dessous le r´esultat principal de cet article.
Th´eor`eme. Soit Γ = P SL(2, Z), alors les sous-facteurs correspondants aux op´erateurs de Hecke sont irr´eductibles et ont comme indice un nombre entier. Les commutants relatifs sup´erieurs sont engendr´es par les projecteurs de Jones. Par cons´equente le sous-facteur est de type A∞. La construction analogue pour Γ = P SL(3, Z) a
comme graphe principal ([Oc], [Po1]) un arbre infini. Par cons´equente, L(P SL(3, Z)) a un sous-facteur d’indice fini et de profondeur infinie.
In this paper we show that the arithmetic Hecke operators acting on P SL(2, Z)-invariant functions on the upper halfplane, are completely positive maps for the mul-tiplicative structure given by the P SL(2, Z)-invariant Berezin deformation quanti-zation of the upper half plane. The structure of this algebra may be determined by the computations of the von Neumann dimensions in ([GHJ], [Ra2]) for the left L(P SL(2, Z))-Hilbert modules defined by the restriction of unitary representations of P SL(2, R) to P SL(2, Z).
The Hecke operators are a sum over a set of cosets and each of these summands will be a completely positive map. Each of these completely positive maps is conse-quently a matrix coefficient for some embedding of the above mentioned Berezin’s quantization algebra into a tensor product of the algebra itself with Mn(C) for some
n, depending on the coset.
By Connes’ work on correspondences, each completely positive map determines a subfactor. We compute the invariants coming from the Jones’ basic construction for the subfactors obtained by the above method and show that each subfactor has as index the square of an integer, depending on the coset. We will show that the fusion algebra is a quotient of a subalgebra of the Hecke algebra of double cosets. The Hilbert bimodule is simply in this case the l2 space l2(ΓαΓ) associated with a
double coset ΓαΓ of Γ.
This type of finite index correspondences may be in fact constructed whenever we are given an almost normal subgroup of a discrete group. Consequently,
an-other interesting example could possibly be derived from the almost normal groups considered in [BC].
For Γ = P SL(2, Z) the corresponding subfactors are irreducible and the higher relative commutants are generated by the Jones’ projections. Hence the associated graph is of type A∞. A similar construction for Γ = P SL(3, Z) will have as
prin-cipal graph ([Oc], [Po1]) an infinite tree. Consequently, L(P SL(3, Z)) has a finite index subfactor of infinite depth (a fact that can not be derived directly from the construction of D. Bisch and U. Haagerup ([BH])).
First we describe an extension of the Hecke operators to the linear space of intertwiners between representations of Γ = P SL(2, Z) that are obtained from the restriction to Γ of the representations in the analytic, discrete series, of unitary representations of P SL(2, R).
Let πn, n ≥ 2 be the discrete series of unitary representation of P SL(2, R) on
the Hilbert spaces Hn = H2(H, yn−2dxdy) where H is the upper part of the
com-plex plane. By restricting this representations to Γ one obtains that Hn are left
Hilbert modules over the von Neumann algebra L(Γ) (see [AS], [Co3], [GHJ]). The corresponding von Neumann dimension ([GHJ]) of this left Hilbert modules over L(P SL(2, Z)) is (n − 1)/12.
Let IntΓ(Hn, Hm) be the set of all linear operators A : Hn → Hm that are
intertwiners for the two representations πn|Γ and πm|Γ (i. e. Aπn(γ) = πm(γ)A
for all γ in Γ). Let f be an automorphic form of weight k. As it was proved in [GHJ] the linear operator of multiplication with f is a bounded operator from Hn
into Hn+k which intertwines the representations πn|Γ and πn+k|Γ.
The arithmetic Hecke operators act linearly on the space of automorphic forms and thus on a subspace of the set of intertwiners. In the next definition we will
show that the action of the Hecke operators can be canonically extended to the linear space of all intertwiners in IntΓ(Hn, Hn+k).
Definition. Fix α in GL(2, Q)/{±1} and let (γi)ki=1 be a system of representatives
for the cosets of Γ/(Γ ∩ αΓα−1). This system is always finite as Γ is an almost nor-mal subgroup of GL(2, Q)/{±1}. By [Ko] we have ΓαΓ = ∪Γαγi. The arguments
in [Ko], for the case of automorphic forms, prove that the following linear map Φα
on IntΓ(Hn, Hm) is actually an endomorphism. The formula for Φα is
Φα(A) = 1 k k ! i=1 πm((αγi)−1)Aπn(αγi).
In particular, if n = m, then Φα defines a completely positive map on IntΓ(Hn, Hn),
i. e. on the commutant {πn(Γ)}′ (which is a type II1 factor). Note that the Hecke
operators are a sum of the Φα’s over a set of representatives of double Γ-cosets.
Remark. For any bounded Γ− invariant function f on H let Tn
f be the Toeplitz
operator acting on the space of analytic functions Hn with symbol f . In [Ra2] it was
proved that {πn(Γ)}′ is the weak closure of the Toeplitz operators with Γ-invariant
symbol. With the above identification, the map Φα will be given by the formula:
Φαf (z) = 1 n ! i f ((αγi)−1z), z ∈ H. In particular"
det α=kΦα is the usual Hecke operator (commuting with the invariant
Laplacian) acting on Γ-invariant functions (see e.g. [Za1] for this definition of the Hecke operators).
The main result of this paper is the identification of the subfactor associated with Φα via Connes’ correspondence construction. Recall from [GHJ] that in this
case dimΓH13 = 1 so that in this case A13 = {π13(Γ)}′ is isomorphic to the group
von Neumann algebra L(Γ).
Theorem. For r ≥ 13, r ∈ N, let Ar = {πr(Γ)}′′ ⊆ B(Hr). Fix α in GL(2, Q)/{±1}
and let (γi)ni=1 be a system of representatives for the cosets of Γ/(Γ ∩ αΓα −1
). For A in Ar, let Φα(A) = (1/n)"ni=1πr((αγi)−1)Aπr(αγi).
Let Hα be the completion of the tensor product Ar⊗ Ar with respect to the norm
given by the scalar product < x ⊗ y, x ⊗ y >= τAr(y
∗
Φα(x∗x)y), for x, y ∈ Ar. We
let Ar act on the left and on the right on Hα by multiplication ([Co1]). Then Hα
is the Connes’ correspondence for the completely positive map Φα.
If r=13, then by identifying {πr(Γ)}′ with L(Γ) (which is possible by [GHJ]),
the correspondence Hα is isomorphic with the L(Γ) bimodule l2(ΓαΓ), with the
canonical action on the left and on the right of Γ by multiplication. If r > 13 then Hα is obtained from the correspondence l2(ΓαΓ) by reduction by a projection.
Proof. Since dimΓ(Hr) ≥ 1, (by [GHJ]), there exists a vector ζ in Hr which is a
separating trace vector in Hr. Hence, for any x, y ∈ Ar, we have
τAr(y
∗
Φα(x∗x)y) = ⟨y∗Φα(x∗x)yζ, ζ⟩ =
!
i
⟨Ui∗x∗xUiyζ, yζ⟩ =
!
i
||xUiyζ||2τ.
Hence, as a Ar-bimodule, Hα is identified with the weak closure
(1) K = Spanw{⊕ixUiyζ|x, y ∈ Ar} ⊆ ⊕iHr.
The left action of Ar is simply left diagonal multiplication by the elements in Ar.
The right action of z in Ar is described by
We will show later that in fact the subset relation in (1) is an equality relation. The algebra of right multiplication operators with elements in Ar is equal to the
weak closure of the linear span {Rγ|γ ∈ Γ} ⊆ B(Hrn), where Rγ for γ ∈ Γ is defined
by
⊕ixUiyζ → ⊕ixUiπr(γ)yζ.
The commutant of the left multiplication operators is clearly identified with Mn(C) ⊗ {πr(Γ)}′′. Here, the Hilbert space on which Mn(C) acts is identified with
l2(Γ/(Γ ∩ αΓα−1
)).
The operators Rγ are identified with elements in Mn(C)⊗{πr(Γ)}′′ in the
follow-ing way. Let φ(γ) be the operator of right multiplication by γ on l2(Γ/Γ ∩ aΓα−1
). Then Rγ is identified with the restriction to K of (⊕Ui)(πr(γ) ⊗ φ(γ)(⊕Ui)∗).
If n=13 and K = Hn
13, then it is clear that the subfactor realized by considering
the left multiplication operators as a subset of the commutant of the right multi-plication operators is, because Ui = π13(αγi), the same as the subfactor coming by
embedding the left multiplication operators into the commutant of the right mul-tiplication operators in the L(Γ)-bimodule l2(ΓαΓ). If we show that the subfactor corresponding to the L(Γ)-bimodule l2(ΓαΓ) is irreducible, then it will follow au-tomatically that K = Hn
13. This will be proved below. Otherwise, if r > 13, then
our argument shows that the subfactor associated with the correspondence is the weak closure
Spanw{(⊕Ui)(πr(γ) ⊗ φ(γ)(⊕Ui) ∗
|γ ∈ Γ} ⊆ Mn(C) ⊗ {πr(Γ)} ′′
.
This last subfactor is obtained from the one corresponding to l2(ΓαΓ), after suitable identifications, by applying πr.
Corollary. Let α be an element in GL(2, Q)/{±1}. Let Γ0 = Γ ∩ αΓα−1. Let
(Γ0γi)i=1,...,n be a system of representatives for Γ/Γ0. Let Φα be the (arithmetic)
linear operator defined on Γ-invariant functions by
Φα(f )(z) =
!
i
f ((αγi)−1z), for z ∈ H.
Let ∗r be the associative (Berezin) product on the Γ-invariant functions on H
that is obtained by using the composition rule for Toeplitz operators on Hr =
H2(H, yr−2dxdy) having symbols Γ−invariant functions. Let Ar be the von
Neu-mann algebra which is obtained via the G. N. S. construction by using the above product ∗r. The trace on this algebra of functions is the integral over a fundamental
domain for Γ in H.
Assume r ≥ 13. Then there exist an embedding Ψα = (Ψa)i,j=1,...,n : Ar →
Ar⊗ Mn(C) whose 1-1 entry (Ψa)1,1 is equal to Φα. Moreover the subfactor
corre-sponding to the embedding Ψa has index n2, is irreducible and of infinite depth.
Note that the usual arithmetic Hecke operators are a sum of Φα when α runs in
a system of representatives of 2 by 2 matrices of determinant n. The properties of the associated subfactor will be proved below.
Proposition. Let α be an element in GL(2, Q)/{±1} and let l2(ΓαΓ) be the L(Γ)-bimodule associated with α. Let n be the index of the subgroup Γ ∩ αΓα−1 in Γ.
Then the subfactor realized as the inclusion of the left multiplication operators into the commutant of the right action of Γ on l2(ΓαΓ) is irreducible, of index n2.
The n-th step in the iterated basic construction of the subfactor corresponds to the bimodule l2(ΓαΓα−1
Γ....α±1
The product involves n successive occurrences of either α or α−1. In particular the
fusion algebra of this subfactor is the subalgebra of the Hecke algebra of double cosets which is generated by ΓαΓ and Γα−1
Γ, modulo scalar multiples of the identity.
Proof. Let Γ0 = Γ ∩ αΓα−1. To compute the index of the associated subfactor
we let (γiΓ0)i=1,...,n be a system of coset representatives for Γ/Γ0. Then (see e.g.
[Ko],[Se]) we have ΓαΓ = ∪iΓαγi. Hence, as left L(Γ)-modules, we may identify
l2(ΓαΓ) = ⊕il2(Γαγi) with the direct sum of n copies of l2(Γ).
Let Γ1 be the kernel of the left multiplication map on l2(Γ/Γ0). Since the right
action of Γ1 is unitary equivalent to a diagonal action on the direct sum of the n-th
copies of l2(Γ), it follows that the dimension of l2(ΓαΓ) as a right module over Γ
is equal to n, which is also the dimension for the left action. The result now follows from the following lemmas
Lemma. Let K1, K2 be two Γ double cosets in GL(2, Q)/{±1}. The Connes’
ten-sor product of the two L(Γ) correspondences over L(Γ) is isomorphic to the Hilbert module l2(K
1K2) with the corresponding left and right action of Γ.
Lemma. Let α, β be two elements in GL(2, Q)/{±1} and assume that l2(ΓαΓ) and
l2(ΓβΓ) are isomorphic as L(Γ)-bimodules (correspondences). Then the two cosets ΓαΓ and ΓβΓ differ only by a rational multiple of the identity.
Proof. Assume that the two bimodules are isomorphic. Let ζ in l2(ΓαΓ) be the image of α under this isomorphism. Then we must have γζ = ζα−1
γα for all γ ∈ Γ ∩ αΓα−1
, and hence η = ζα−1
must simultaneously belong to l2(ΓβΓα−1
) and to the commutant of Γ ∩ αΓα−1
Lemma. The conjugate correspondence of l2(ΓαΓ) is l2(Γα−1Γ).
The theorem now follows because the iterated term in the Jones’ basic con-struction of a subfactor given by a correspondence H over a type II1 factor M is
obtained by recursively taking the tensor product of the initial correspondence with its opposite. This also completes the proof of our first theorem, because it shows that the factor associated to Φα is irreducible (since we have proven that equality
holds in the relation (1)).
Corollary. Let α be given by the matrix # p 0 0 1
$
in GL(2, Q)/±1. Let Γ =
P SL(2, Z). Then the subfactor associated with the correspondence l2(ΓαΓ) is
irre-ducible and has index n2 where n is the group index [Γ : Γ ∩ αΓα−1]. Moreover the
higher relative commutants are generated only by the corresponding Jones’ projec-tions. In particular the graph of the subfactor is A∞. The similar construction for
P SL(3, Z) instead of P SL(2, Z) will yield a subfactor of integer index and whose graph is an infinite tree.
Proof. We use the double coset computations for the Hecke algebra as explained in the book of Andrianov (see also [Shi]). Because our computations are modulo the scalar multiples of the identity, the bimodule for the coset Γα−1
Γ is the same as for the coset ΓαΓ. The algebra of cosets for diagonal elements in GL(2, Q)/{±1} having only powers of p on the diagonal is isomorphic with the algebra of symmetric polynomials in two variables x1, x2 ([An]). Set Xk = l2(ΓαkΓ) as a L(Γ) bimodule.
Then, via the above isomorphism Xk is identified with the polynomial xk1 + xk2.
Hence Xk⊗L(Γ)X1 is identified with (xk+11 + x2k+1) +(xk1x2+ xk2x1). The second
summand corresponds to Γ# p
k 0
0 p
$
last bimodule will give the same correspondence as Γαk−1Γ. By what we have just proved and by [An] it follows that Xk is an irreducible bimodule and that
Xk⊗L(Γ)X1 is isomorphic (as a bimodule) to Xk+1⊕ Xk−1.
The computations for P SL(3, Z) are similar.
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Permanent Address: Department of Mathematics, The University of Iowa, Iowa City, IA 52246, U.S.A.
Member of the Institute of Mathematics Romanian Academy, Bucharest, Roma-nia
